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Abstract
The optical fiber intrusion signal detection technology adopts the distributed optical fiber as sensor to monitor and identify perimeter intrusion signals. Due to the diversity of perimeter intrusion, the optical fiber intrusion signals are composed of various types of pure signals. Therefore, direct identification on mixed signals will cause the system performance degradation. A more effective method is to first unmix the mixed signal to obtain each pure signal component, and then perform signal identification. In this paper, nonorthogonal principal skewness analysis (NPSA) based unmixing algorithm is proposed. By introducing supersymmetric tensors, the mixed fiber signal unmixing problem is transformed into the skewness analysis problem, and then the non-orthogonal solution is further solved which significantly improves the accuracy of solving pure signal components. The effectiveness of the proposed algorithm is verified by actual data experiments.
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1. Introduction
The optical fiber pre-warning system (OFPS) is widely used in the detection and identification of optical fiber intrusion signals [1], such as the detection of pipeline leakage, the protection of airports and military bases, and the identification of human and mechanical intrusion behavior [2]. The OFPS uses underground cables to detect and identify external intrusion signals, thereby calculating the location and time of any abnormal events. The main principle is that the intrusion behavior causes weak vibration at the corresponding position of the fiber, which changes the refractive index of the corresponding position of the fiber. There are various intrusion behaviors, and their power spectrum curves are also different. The power spectrum calibrated by the laboratory is called the characteristic power spectrum [3] which is easier to recognize compared with mixed signals. In reality, the intrusion behavior is usually accompanied by various factors, and the intrusion signal is actually a mixture of various pure signals. Therefore, the spectrum of intrusion signal will be a mixed spectrum. In order to accurately analyze the type of intrusion signal, the OFPS system must have unmixing capabilities.
There are many algorithms for fiber unmixing. In the traditional algorithm, Y. Shi et al. studied the purpose of unmixing fiber signals by setting an appropriate wavelet threshold by comprehensively considering the fiber noise power [4,5]. However, this hard threshold denoising method cannot overcome the influence of signal jitter and optical path variation. In the face of uniform background noise, the above methods have achieved good results, but in the face of non-uniform background noise environment, it is difficult to achieve good detection results using a fixed threshold. HQ Qu et al. proposed a method for unmixing optical fiber intrusion signals based on the constant false alarm (CFAR) algorithm [6,7]. By adjusting the threshold value adaptively to different background noises, it greatly improves the detection of optical fiber intrusion signals. At present, the unmixing method based on neural network has gradually become a research hotspot. We usually assume that the mixed power spectrum of OFPS is a linear superposition of the characteristic spectrum, and the proportion of each component is distributed the interval [0,1] [8]. From the statistical perspective, the mixed fiber signal samples are distributed in the simplex with the characteristic spectrum as the endpoint. Therefore, the essence of OFPS unmixing is to find the simplex endpoint where the sample data is located. Based on this model, two unmixing methods are proposed. The first one takes the simplex as the objective function and solves it by the mathematical optimization method, which is called the SVA algorithm [8–10]. The SVA algorithm has no requirements on the analyticity of the objective function and is widely applicable. The second is based on matrix decomposition to perform blind separation of mixed signals. A related typical algorithm is non-negative matrix factorization (NMF). The NMF algorithm uses the non-negativity of the matrix to achieve signal decomposition, which belongs to the unsupervised learning algorithm. The algorithm has strong physical properties, simple and fast calculation, and does not require the independence and sparsity of the decomposed signal. However, the problem with these two methods is that the number of characteristic power spectrum in samples needs to be known, which is unrealistic in actual processing. Because the optical fiber transmission usually takes tens of kilometers with extreme random intrusion behavior, it is difficult to know the number of intrusion signal types. To solve this problem, the principal component analysis (PCA) is adopted [11–13]. The PCA mainly focus on the second-order statistical characteristics of the samples, which cannot reveal the intrinsic structure of the intrusion signals. Considering the above problem, we propose the nonorthogonal principal skewness analysis (NPSA) based on matrix theory, which introduces supersymmetric tensors for intrusion signals and transforms the problem of solving the local maximum skewness direction into the problem of solving the eigenvalues and eigenvectors of the residual tensors. First, initial eigenvalues and eigenvectors are obtained by means of fixed-point method. Considering the inherent non-orthogonality of supersymmetric tensors, this algorithm uses the orthogonal complement of the Kronecker product of the previous vector and its own vector to expand the search space of eigenvectors instead of the orthogonal complement space. After repeated iterations, the new eigenvalues and eigenvectors are obtained finally, which correspond to the pure signals. The experimental results show that this algorithm can get more accurate unmixing effect.
The organization structure of this paper is as follows. Section 2 introduces the unmixing model, and then Section 3 introduces the NPSA algorithm in detail. The experimental analysis is given in Section 4. Finally, the conclusion is provided in Section 5.
2. Unmixing model
In OFPS, the optical fiber is usually buried under the ground of the monitored object, and distributed optical fiber is used as the sensor and transmission channel to collect and analyze the vibration data. The vibration data acquisition is based on φ-OTDR principle. If there is any form of vibration above them, it will be propagated to the corresponding position of optical fiber and then modulate the backscattered optical fiber signal. If there are various types of vibration effects at the same time, the backscattered optical fiber signals will be modulated by superimposed vibration. And then goes through amplifier, analog/digital converters (ADC) to extract vibration data. After that, the vibration data is detected by the detection unit [5,7], and sent to the unmixing unit to achieve signal unmixing. The algorithm in this paper is used in the unmixing unit, and finally the corresponding type is identified by the recognition unit [14]. OFPS system data acquisition and signal processing flow is shown in Fig. 1.Here, the power spectrum is taken as the feature of the signal [15,16]. The process flow of power spectrum extraction of the optical fiber intrusion signal is shown in Fig. 2. Firstly, the time-frequency analysis is performed on the optical fiber intrusion signal to obtain the time-frequency distribution at each spatial position, on which energy integration is carried out band-wise along the frequency direction to generate the spatial-temporal two-dimensional images in each frequency band. Then, these two-dimensional images corresponding to each frequency band are further stacked up to form a spatial-temporal-frequency three-dimensional cube data, based on which we finally extract the power spectrum as the feature of the analyzed optical fiber intrusion signal.
Assuming that the number of mixed fiber sample signals is N, the fiber sample signals is formed by linear superposition of K pure intrusion signal samples, ${e_j}$ is the power spectrum corresponding to j-th intrusion signal, where j = 1,2,3…, K. The i-th observed power spectrum is expressed as
where E is the $L \times K$ order characteristic spectrum matrix, i.e., ${\textbf E} = [{e_1},{e_2},\ldots ,{e_k}]$, n is additive noise, ${a_i}$ is the regularization coefficient, ${a_i} = {[{a_{1i}},{a_{2i}}, \cdots ,{a_{Ki}}]^T}$ which should be satisfied byIn the OFPS unmixing model, the purpose is to obtain the eigenmatrix E, which is obtained by solving the eigenvector.
3. Unmixing algorithm based on principal skewness analysis
The main idea of the algorithm is to transform the unmixing problem into principal skewness analysis problem. We introduce the supersymmetric tensor S and considers the non-orthogonality inherent of the eigenvector of the supersymmetric tensor, which uses the orthogonal complement of the Kronecker product of the former eigenvector and itself to enlarge the search space of the eigenvector. The specific calculation steps are as follows.
Firstly, we calculate the initial eigenvector E. Assuming that the optical fiber signal is ri, the supersymmetric tensor S is first obtained through the cross-product operation of the optical fiber signal as is given by
where N is the sample number of optical fiber signal, and ${\circ}$ is the cross product of two signals. Then, the initial eigenvector E can be obtained byThen, we replace the tensor S with vector s through vectorization.
where, $\mathop s\nolimits_i \in \mathop R\nolimits^{\mathop L\nolimits^2 \times 1}$,L is the dimension of the sample, L2 means that the dimension of the sample is 2, $s \in \mathop R\nolimits^{\mathop L\nolimits^3 \times 1}$, L3 means that the dimension of the sample is 3, and vec means that the tensor is changed into a vector. Specifically, the tensor s is first expanded into a two-dimensional plane, and then according to formula (6), the two-dimensional plane is expanded into a one-dimensional vector.Next, a new eigenvector is obtained by calculating the third Kronecker product of eigenvector E.
The orthogonal complement projection matrix of the new eigenvector is calculated by
A new higher-order tensor can be obtained through multiplying the orthogonal complement projection matrix of the new eigenvectors by the vector s as expressed below,
where unvec means vector transform tensor.Finally, the updated tensor S is used to obtain the final eigenvector E, that is, the unmixed pure signal. The optimization model expression is as follows, and λ is regularization factor:
After several iterations, we can obtain the final eigenvector.
In this paper, we use the simplex to measure the performance of the algorithm as shown in Fig. 3. The smaller the simplex volume surrounded by the pure signal, the more accurate the estimate. The simplex volume is represented by the following equation
4. Unmixing experiments
4.1 Introduction to the data sets
The optical fiber signal data used in this experiment is collected from the experimental site in the suburbs of Beijing. The optical fiber is buried 20-30cm below the surface. The optical fiber pre-warning system is mainly composed of computer case, optical transmitting and receiving board, data acquisition board and so on, which is shown in Fig. 4. Among various vibration source types, mechanical intrusion has the highest hazard and waveform regularity, showing the characteristics of high frequency and periodicity. The most common mechanical signals are the signals of electric drills, electric picks, and picks. Therefore, these three signals are selected as typical pure signals in this paper. The characteristic power spectrum corresponding to these three pure intrusion signals are shown in Fig. 5.
Fig. 4. The system component: (a) computer case (b) optical transmitting and receiving board (c) data acquisition board
Fig. 5. The characteristic power spectrum of three pure intrusion signals: (a) electric pick signal; (b) electric drill signal;(c) digging signal
It can be seen from Fig. 5 that the characteristic spectrum of the electric pick signal has a peak response at about 60 Hz and the spectrum of the electric drill signal has multiple responses at low frequencies with an extreme point appears when the frequency is around 60 Hz. The frequency spectrum of digging signal has a peak response at the frequency of 20 Hz. In the subsequent unmixing experiment, these three intrusion behaviors will be operated at the same time to generate the mixed intrusion signals.
4.2 Unmixing experiment
In this section, we use this algorithm to extract pure intrusion fiber signals from the above data sets. For intuitive show, we project the sample data onto a two-dimensional plane as shown in Fig. 6. The blue dots represent mixed fiber intrusion signal samples, while the red dots represent pure intrusion signals extracted by this method. In this two-dimensional plane, the sample data are randomly distributed in a triangle composed of three pure intrusion signals.
According to formula (10), we use different λ values to perform pure signal extraction. In the third subsection, the concept of simplex is introduced, and the λ parameter limits the contribution of the simplex volume to the eigenvectors. The smaller the λ value is, the smaller the enclosed volume is, and the demixed fiber signal is closer to the pure signal. The region surrounded by the extracted pure signal in Fig. 6(a) and (b) is smaller than the region surrounded by mixed data points, so the extraction in both cases is invalid. In Fig. 6(d), compared with Fig. 6(c), the extracted pure signal is closer to the end of the triangle, and the region surrounded by the pure signal happens to contain mixed data points. Therefore, the effect of pure signal extraction is more accurate when λ=0.001.
After unmixing process, the signal recognition is carried out. The three pure power spectra extracted under different λ values are shown in Fig. 7. The black line represents the characteristic spectrum, that is, the pure signal spectrum after laboratory calibration. The red, green, yellow and blue lines represent the case with λ=0.001, 0.01, 0.05 and 0.1 in turn. It can be seen from Fig. 7 that the pure spectrum extracted at λ=0.001 is closer to the characteristic spectrum.
In order to evaluate the unmixing effect under different λ values more precisely, the mean square error (MSE) is used to calculate the unmixing error in this experiment, which is defined as follows,
The smaller the MSE value, the closer the predicted value is to the true value. The MSE at different λ values are shown in Table 1. According to the table, when λ=0.001, the MSE has the smallest value indicating that the pure intrusion signal at this time is very close to the actual pure intrusion signal.
Table 1. The MSE of unmixing results under different λ values
4.3 Comparison with other algorithms
In this section, we use NMF as the reference algorithm to illustrate the advantages of this approach. The principle of the NMF algorithm can be described as: under the non-negative condition, the given matrix is decomposed into the product of two sub-non-negative matrices, and its mathematical expression can be expressed as:
can also be written as:Among them, ${\textbf R} \in {I^{m \times N}}$ represents the observed non-negative matrix, which can be denoted as ${\textbf R} = [{{\textbf r}_1},{{\textbf r}_2},\ldots ,{{\textbf r}_N}]$;${\textbf E} \in {I^{m \times n}}$ is called the basis matrix, which can be regarded as a mixture matrix in the blind signal separation model;${\textbf A} \in {I^{n \times N}}$ is called the coefficient matrix, which can also be regarded as the recovered source signal matrix, denoted as;${\textbf A} = [{{\textbf a}_1},{{\textbf a}_2},\ldots ,{{\textbf a}_N}]$;${{\textbf r}_t}$ and ${{\textbf a}_t}$ respectively represent the t-th column vector of the non-negative matrix sum R and A. ${{\textbf r}_t}$ can be regarded as the weighted sum of all E column vectors, ${{\textbf a}_t}$ is the weighting coefficient. The NMF problem is actually a problem of minimizing the error between R and EA under the non-negative constraints of E and A, where the constraint is that all elements in E and A must be greater than 0.”
The unmixing results of NMF algorithm are shown in Fig. 8(b). It can be clearly seen from Fig. 8, the simplex volume surrounded by NMF is much larger than the mixed data region, which degrades the unmixing effect.
The pure power spectrum extracted by the two algorithms is shown in Fig. 9. The black line represents the characteristic power spectrum, and the red line and blue line represent the unmixing results of NPSA and NMF, respectively. It can be seen from Fig. 9, the pure spectrum extracted by the algorithm in this paper is much closer to the characteristic spectrum than that extracted by the NMF algorithm. In addition, the MSE obtained by the two methods is shown in Table 2. The MSE of this algorithm is much smaller indicating that this algorithm has obvious advantages in unmixing.
Table 2. MSE values of the two algorithms
5. Conclusion
In this paper, the basic framework and unmixing model of optical fiber warning system are briefly introduced. The simplex concept is introduced to measure the unmixing effect of mixed signals. According to the experiment, under the condition of λ=0.001, the extracted pure signal is closer to the end of the triangle, and the area surrounded by the pure signal just contains mixed data points. Under this condition, the mean square error is the smallest, and the power spectrum of the pure signal is closer to the characteristic spectrum. All above results indicate that λ=0.001 has the best unmixing effect compared with other λ values. Further comparison with NMF algorithm shows that the proposed algorithm is better than the NMF algorithm with more accurate unmixing effect.
Funding
National Key Research and Development Program of China (2020YFB1600700); Beijing Municipal Natural Science Foundation (4202019).
Disclosures
There are no financial conflicts of interest to disclose.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author, [Linjing Jia], upon reasonable request.
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